A361804 Number of partitions of [n] with an equal number of even and odd block sizes.

1, 0, 0, 3, 0, 15, 45, 63, 1260, 1515, 25515, 104973, 510345, 5679765, 17252235, 263214318, 1207222380, 11863296915, 101718989235, 630468648873, 8281982665215, 48583038314415, 656006633919945, 5122900223419938, 54304561161840825, 605082149235374265

Offset: 0

Author

Alois P. Heinz, Jun 12 2023

Keywords

• nonn

Comments

Half the number of block sizes are even and the other half are odd.

Examples

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 0.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 0.
a(5) = 15: 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 45: 12|34|5|6, 12|35|4|6, 12|3|45|6, 12|36|4|5, 12|3|46|5, 12|3|4|56, 13|24|5|6, 13|25|4|6, 13|2|45|6, 13|26|4|5, 13|2|46|5, 13|2|4|56, 14|23|5|6, 15|23|4|6, 1|23|45|6, 16|23|4|5, 1|23|46|5, 1|23|4|56, 14|25|3|6, 14|2|35|6, 14|26|3|5, 14|2|36|5, 14|2|3|56, 15|24|3|6, 1|24|35|6, 16|24|3|5, 1|24|36|5, 1|24|3|56, 15|2|34|6, 1|25|34|6, 16|2|34|5, 1|26|34|5, 1|2|34|56, 15|26|3|4, 15|2|36|4, 15|2|3|46, 16|25|3|4, 1|25|36|4, 1|25|3|46, 16|2|35|4, 1|26|35|4, 1|2|35|46, 16|2|3|45, 1|26|3|45, 1|2|36|45.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A003724, A004767, A004773, A005046, A275679.

Programs

Maple

b:= proc(n, x, y) option remember; `if`(abs(x-y)>2*n, 0,
     `if`(n=0, 1, b(n-1, x+1, y)+`if`(x>0, b(n-1, x-1, y+1)*x, 0)+
     `if`(y>0, b(n-1, x+1, y-1)*y, 0)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..33);

Formula

a(n) mod 5 = 3 for n in { A004767 }, a(n) mod 5 = 1 for n = 0 and a(n) mod 5 = 0 for all other n (n in { A004773 } \ { 0 }).

a(n) mod 3 = 0 for n >= 1.